The Annals of Statistics

Conditional mean and quantile dependence testing in high dimension

Xianyang Zhang, Shun Yao, and Xiaofeng Shao

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Motivated by applications in biological science, we propose a novel test to assess the conditional mean dependence of a response variable on a large number of covariates. Our procedure is built on the martingale difference divergence recently proposed in Shao and Zhang [J. Amer. Statist. Assoc. 109 (2014) 1302–1318], and it is able to detect certain type of departure from the null hypothesis of conditional mean independence without making any specific model assumptions. Theoretically, we establish the asymptotic normality of the proposed test statistic under suitable assumption on the eigenvalues of a Hermitian operator, which is constructed based on the characteristic function of the covariates. These conditions can be simplified under banded dependence structure on the covariates or Gaussian design. To account for heterogeneity within the data, we further develop a testing procedure for conditional quantile independence at a given quantile level and provide an asymptotic justification. Empirically, our test of conditional mean independence delivers comparable results to the competitor, which was constructed under the linear model framework, when the underlying model is linear. It significantly outperforms the competitor when the conditional mean admits a nonlinear form.

Article information

Ann. Statist., Volume 46, Number 1 (2018), 219-246.

Received: January 2016
Revised: November 2016
First available in Project Euclid: 22 February 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Large-$p$-small-$n$ martingale difference divergence simultaneous test $U$-statistics


Zhang, Xianyang; Yao, Shun; Shao, Xiaofeng. Conditional mean and quantile dependence testing in high dimension. Ann. Statist. 46 (2018), no. 1, 219--246. doi:10.1214/17-AOS1548.

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Supplemental materials

  • Supplement to “Conditional mean and quantile dependence testing in high dimension”. This supplement contains proofs of the main results in the paper, extension to factorial designs, additional discussions and numerical results.